Year 7 Mathematics · AC9M7P02

Repeated Experiments and Probability Simulation

conduct repeated chance experiments and run simulations with a large number of trials using digital tools; compare predictions about outcomes with observed results…

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Learning goalsSay it simply

We are learning to conduct repeated chance experiments and digital simulations and compare observed with expected frequencies.

Relative frequency is the observed event count divided by the total number of trials, so it allows results from different sample sizes to be compared fairly. Expected frequency is theoretical probability multiplied by trial count, and the difference between observed and expected counts records what happened in one run rather than automatically proving an error.

Random variation means repeated samples from the same valid chance process can produce different proportions. As the number of independent trials grows, relative frequency usually becomes more stable around theoretical probability, but it need not approach the target steadily or equal it exactly at any particular trial count.

A digital simulation must reproduce the intended outcome probabilities, generate trials independently where required, record every valid result and disclose its trial count and settings. A small manual trace, total-count check and comparison across repeated runs help detect coding, weighting or recording errors.

Success criteria

  • I can calculate and compare relative frequency, theoretical probability and expected frequency.
  • I can explain random variation and why larger samples usually give more stable estimates.
  • I can design or audit a reproducible simulation so its random process matches the intended probability model.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Comparing observed and expected coin-toss frequencies.

  1. A fair-coin simulation records 103 heads in 200 independent tosses.
  2. Calculate the observed relative frequency as 103 divided by 200, which equals 0.515.
  3. Calculate the expected heads frequency as 200 × 0.5 = 100 and the observed-minus-expected difference as 103 − 100 = 3.
  4. Conclude that 0.515 is close to the theoretical probability 0.5 and that three extra heads are plausible random variation rather than evidence that the coin is unfair.

Final answer: The observed relative frequency is 0.515, compared with theoretical probability 0.5 and expected frequency 100 heads, so the difference is 3 heads.

Check: The observed counts 103 heads and 97 tails total all 200 trials, and 0.515 + 0.485 = 1.

Example 2

Designing and interpreting a 30-percent simulation.

  1. Generate a random integer from 1 to 10 independently on each trial and define success as an output of 1, 2 or 3, giving theoretical probability 3/10 = 0.3.
  2. Run 500 trials and record 163 successes, then calculate relative frequency as 163/500 = 0.326.
  3. Calculate expected frequency as 500 × 0.3 = 150 and compare the observed count, which is 13 above expected.
  4. Repeat the 500-trial simulation and inspect the generator settings before deciding whether 0.326 reflects ordinary variation or a persistent model problem.

Final answer: The simulation has intended probability 0.3, expected frequency 150 and observed relative frequency 0.326 from 163 successes.

Check: Three of the ten equally likely generator outputs are successes, all 500 trials are counted, and repeated runs can test whether the difference persists.

Example 3

Application problem 1

Problem: A fair coin simulation gives 7 heads in 10 trials and 503 heads in 1000 trials. Compare both relative frequencies with the theoretical probability and explain the difference.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: The relative frequencies are 0.7 and 0.503, compared with theoretical 0.5. Larger samples usually show less relative variation, but an exact 0.5 is not guaranteed.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: The relative frequencies are 0.7 and 0.503, compared with theoretical 0.5. Larger samples usually show less relative variation, but an exact 0.5 is not guaranteed.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Plan a reproducible 1,000-trial simulation for a spinner with probabilities red 1/2, blue 1/4 and green 1/4, including two validity checks and the summaries to report.

  1. Plan: Specify equal random outputs, map the correct number to each colour and state how totals and expected values will be checked.
  2. Work: Generate independent integers 1 to 4, map 1 and 2 to red, 3 to blue and 4 to green, run exactly 1,000 trials, check that every output is recorded and that the mapping gives probabilities 2/4, 1/4 and 1/4, then report each count, relative frequency, expected frequency and observed-minus-expected difference across repeated runs.
  3. Interpret: The plan aligns generator weights with the spinner and makes both implementation and results auditable.

Final answer: Generate independent integers 1 to 4, map 1 and 2 to red, 3 to blue and 4 to green, run exactly 1,000 trials, check that every output is recorded and that the mapping gives probabilities 2/4, 1/4 and 1/4, then report each count, relative frequency, expected frequency and observed-minus-expected difference across repeated runs.

Check: The plan aligns generator weights with the spinner and makes both implementation and results auditable.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7P02 - Repeated Experiments and Probability Simulation
Example 1

Example 1 Comparing observed and expected coin-toss frequencies. A fair-coin simulation records 103 heads in 200 independent tosses. Calculate the observed relative frequency as 103 divided by 200, which equals 0.515. Calculate the expected heads frequency as 200 × 0.5 = 100 and the observed-minus-expected difference as 103 − 100 = 3. Conclude that 0.515 is close to the theoretical probability 0.5 and that three extra heads are plausible random variation rather than evidence that the coin is unfair. Final answer: The observed relative frequency is 0.515, compared with theoretical probability 0.5 and expected frequency 100 heads, so the difference is 3 heads. Check: The observed counts 103 heads and 97 tails total all 200 trials, and 0.515 + 0.485 = 1.

Example 2

Example 2 Designing and interpreting a 30-percent simulation. Generate a random integer from 1 to 10 independently on each trial and define success as an output of 1, 2 or 3, giving theoretical probability 3/10 = 0.3. Run 500 trials and record 163 successes, then calculate relative frequency as 163/500 = 0.326. Calculate expected frequency as 500 × 0.3 = 150 and compare the observed count, which is 13 above expected. Repeat the 500-trial simulation and inspect the generator settings before deciding whether 0.326 reflects ordinary variation or a persistent model problem. Final answer: The simulation has intended probability 0.3, expected frequency 150 and observed relative frequency 0.326 from 163 successes. Check: Three of the ten equally likely generator outputs are successes, all 500 trials are counted, and repeated runs can test whether the difference persists.

Example 3

Example 3 Application problem 1 Problem: A fair coin simulation gives 7 heads in 10 trials and 503 heads in 1000 trials. Compare both relative frequencies with the theoretical probability and explain the difference. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: The relative frequencies are 0.7 and 0.503, compared with theoretical 0.5. Larger samples usually show less relative variation, but an exact 0.5 is not guaranteed. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: The relative frequencies are 0.7 and 0.503, compared with theoretical 0.5. Larger samples usually show less relative variation, but an exact 0.5 is not guaranteed. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Plan a reproducible 1,000-trial simulation for a spinner with probabilities red 1/2, blue 1/4 and green 1/4, including two validity checks and the summaries to report. Plan: Specify equal random outputs, map the correct number to each colour and state how totals and expected values will be checked. Work: Generate independent integers 1 to 4, map 1 and 2 to red, 3 to blue and 4 to green, run exactly 1,000 trials, check that every output is recorded and that the mapping gives probabilities 2/4, 1/4 and 1/4, then report each count, relative frequency, expected frequency and observed-minus-expected difference across repeated runs. Interpret: The plan aligns generator weights with the spinner and makes both implementation and results auditable. Final answer: Generate independent integers 1 to 4, map 1 and 2 to red, 3 to blue and 4 to green, run exactly 1,000 trials, check that every output is recorded and that the mapping gives probabilities 2/4, 1/4 and 1/4, then report each count, relative frequency, expected frequency and observed-minus-expected difference across repeated runs. Check: The plan aligns generator weights with the spinner and makes both implementation and results auditable.

Curriculum examplesCopied content

Content description: conduct repeated chance experiments and run simulations with a large number of trials using digital tools; compare predictions about outcomes with observed results, explaining the differences.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. An event occurs 18 times in 30 trials; calculate its relative frequency.

    Check answer

    Answer: The relative frequency is 18/30 = 0.6.

    Hint: Write observed count over total trials and simplify or convert to a decimal.

    Why: Relative frequency divides the observed event count by the total trial count.

  2. 2. An event has theoretical probability 0.4; calculate its expected frequency in 250 trials.

    Check answer

    Answer: The expected frequency is 250 × 0.4 = 100.

    Hint: Multiply 250 by the stated probability rather than using an observed count.

    Why: Expected count is theoretical probability multiplied by the planned number of trials.

  3. 3. Experiment A records 42 successes in 60 trials and Experiment B records 132 successes in 200 trials; which observed relative frequency is greater?

    Check answer

    Answer: Experiment A has relative frequency 42/60 = 0.70, while Experiment B has 132/200 = 0.66, so Experiment A's is greater.

    Hint: Divide each success count by its own trial count before comparing.

    Why: Converting both counts to proportions permits a fair comparison across different totals.

  4. 4. A fair-coin simulation produces 54 heads in 100 tosses; compare observed relative frequency, theoretical probability and expected frequency.

    Check answer

    Answer: The observed relative frequency is 0.54, the theoretical probability is 0.5 and the expected frequency is 50 heads, so the observed count is 4 above expected.

    Hint: Calculate 54/100 for the observation and 100 × 0.5 for the expectation.

    Why: Observed and theoretical values can differ slightly through random variation.

  5. 5. Two runs of the same simulation record 27 successes in 50 trials and 63 successes in 100 trials; find the pooled relative frequency.

    Check answer

    Answer: The pooled relative frequency is (27 + 63)/(50 + 100) = 90/150 = 0.6.

    Hint: Add numerators and denominators separately rather than averaging the two proportions without weights.

    Why: Pooling combines both event counts and both trial counts before division.

  6. 6. Design a random-integer simulation for an event with probability 3/8 and state which outputs represent success.

    Check answer

    Answer: Generate an integer from 1 to 8 with equal probability on every trial and let 1, 2 or 3 represent success.

    Hint: Use eight equal generator outcomes and label exactly three as successes.

    Why: Three successful outputs among eight equally likely outputs model probability 3/8.

  7. 7. A program simulates a fair coin by choosing an integer from 1 to 5 and calling 1, 2 or 3 heads; identify the flaw and repair it.

    Check answer

    Answer: The program gives heads probability 3/5 rather than 1/2, so use an even number of equally likely outputs, such as integers 1 to 10 with 1 to 5 as heads and 6 to 10 as tails.

    Hint: Count the successful generator outputs and compare that fraction with one-half.

    Why: A valid simulation must assign equal total probability to heads and tails.

  8. 8. A fair coin gives 14 heads in 20 tosses and 104 heads in 200 tosses; compare the estimates and explain which is generally more reliable without claiming certainty.

    Check answer

    Answer: The relative frequencies are 0.70 and 0.52, and the 200-toss estimate is generally more reliable because a larger independent sample usually has smaller proportional fluctuations, although no finite run is guaranteed to be closest.

    Hint: Convert both counts to proportions and compare each sample size with the theoretical probability 0.5.

    Why: Sample size affects stability, while chance variation remains present in both runs.

  9. 9. Plan a reproducible 1,000-trial simulation for a spinner with probabilities red 1/2, blue 1/4 and green 1/4, including two validity checks and the summaries to report.

    Check answer

    Answer: Generate independent integers 1 to 4, map 1 and 2 to red, 3 to blue and 4 to green, run exactly 1,000 trials, check that every output is recorded and that the mapping gives probabilities 2/4, 1/4 and 1/4, then report each count, relative frequency, expected frequency and observed-minus-expected difference across repeated runs.

    Hint: Specify equal random outputs, map the correct number to each colour and state how totals and expected values will be checked.

    Why: The plan aligns generator weights with the spinner and makes both implementation and results auditable.

  10. 10. A fair coin simulation gives 7 heads in 10 trials and 503 heads in 1000 trials. Compare both relative frequencies with the theoretical probability and explain the difference.

    Check answer

    Answer: The relative frequencies are 0.7 and 0.503, compared with theoretical 0.5. Larger samples usually show less relative variation, but an exact 0.5 is not guaranteed.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: A fair coin must produce exactly half heads in every set of tosses.

Correction: Half is the theoretical probability, while finite observed proportions vary randomly around it.

Common mistake: Relative frequency must move closer to theoretical probability after every additional trial.

Correction: The estimate can move towards or away on individual steps even though larger samples are usually more stable overall.

Common mistake: A digital random result is automatically valid because a computer produced it.

Correction: The programmed outcomes, weights, independence and recording process must be checked against the intended chance model.

Curriculum alignmentStart here

We are learning to conduct repeated chance experiments and digital simulations and compare observed with expected frequencies.

Relative frequency is the observed event count divided by the total number of trials, so it allows results from different sample sizes to be compared fairly. Expected frequency is theoretical probability multiplied by trial count, and the difference between observed and expected counts records what happened in one run rather than automatically proving an error.

Random variation means repeated samples from the same valid chance process can produce different proportions. As the number of independent trials grows, relative frequency usually becomes more stable around theoretical probability, but it need not approach the target steadily or equal it exactly at any particular trial count.

A digital simulation must reproduce the intended outcome probabilities, generate trials independently where required, record every valid result and disclose its trial count and settings. A small manual trace, total-count check and comparison across repeated runs help detect coding, weighting or recording errors.

Success criteria

  • I can calculate and compare relative frequency, theoretical probability and expected frequency.
  • I can explain random variation and why larger samples usually give more stable estimates.
  • I can design or audit a reproducible simulation so its random process matches the intended probability model.
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